Z-scores, normal distribution, and more

 The bell curve is a symmetric curve, with the center of the graph being the high point, and the two sides on the graph getting shorter and shorter, becoming “tails”  The bell curve is most commonly associated with the normal distribution.  Below is a picture of what a bell curve looks like

 The normal distribution is a continuous distribution that has the bell curve shape.  What if we would like to compare two different normal distributions?  If they have different standard deviations, then it would not be plausible to look at them side by side.  What we can do to get them on the same scale is to standardize the distributions.  This is done by making each point in the distribution into a standard score, or z-score.

For any data point x taken from a normal population with mean and standard deviation the z-score of that point will be: We are subtracting the mean from each point, meaning the distribution is now centered around 0, and dividing by the standard deviation, meaning the distribution now has a standard deviation of 1.

What is the z-score for the point x = 11.5 if it comes from a normal distribution with a mean of 15 and a standard deviation of 2.5? We can plug this into our z-score formula And get z = ( )/2.5 = -3.5/2.5 = -1.4 Our interpretation of this would be that our original data point 11.5 is 1.4 standard deviations below the mean, since our z-score was negative

 If we standard a normal distribution, it is now said to follow a standard normal distribution.  In other words, we have N(, ) to start.  After we standardize it by calculating z-scores, we have the distribution N(0,1)  Our new distribution is normal, with a mean of 0, and a standard deviation of 1.  So why did we want to do this again?

If we have two standard normal distributions, we can compare the two since they are now on the same scale. Another thing we can do is to see how many standard deviations a point lies away from the mean. This is what a z-score tells us. If a z-score is positive, that means the original point was above the mean. If it was negative, then the original point was below the mean. Whatever the value of the number, that is how many standard deviations away from the mean the original value was.

 A continuous random variable is any variable, decided by chance, that can take on any value in a range of numbers. This is most commonly the case with measurements (height, weight, length, time, etc.)  In some cases, continuous random variables follow a normal distribution.  We can graph these variables using the bell curve.

 Suppose the height of NBA players is normally distributed with a mean of 77 inches and a standard deviation of 4 inches.  Using the empirical rule, we can make a graph of this distribution.

The graph is centered around the mean of 77 inches. If we move 1 standard deviation above and below the mean, we will have about 68% of the data between those two points, which are 73 and 81 inches. If we move 2 standard deviations above and below the mean, we will have about 95% of the data between those two points, which are 69 and 85 inches. If we move 3 standard deviations above and below the mean, we will have about 99.7% of the data between those two points, which are 65 and 89 inches.

This is still a bell curve, but now it is centered at 0, with a standard deviation of 1.

 Many variables in life follow the bell curve shape.  A popular example is IQ, which is usually said to have a mean of 100. Obviously, there are very smart people and very… not smart people. But a good percentage of people are somewhere close to the average.  Height is also a good example, but you have to be careful.  If you look at height of males, or of females, then it will follow a bell curve. But if you put them together, it will not be so nice. This is because men and women have different average heights!

This graph shows women’s, in black, and men’s, in red, heights. They are normal on their own. But this graph shows that, put together, it is not as clear as before that it is normal. It almost seems to be more uniform, instead of bell shaped.

 Grades also follow a bell curve. You may have heard that a C is average? Well, this was not just pulled out of the air. This is because this actually is the average grade when you look at all grades. There are plenty of A’s, plenty of F’s, and plenty of C’s, but when you look at the bell curve, we have C in the middle.  Can you think of any other real life applications of the bell curve?